[unix-history] / usr / src / lib / libm / common_source / exp.c

/* 
 * Copyright (c) 1985 Regents of the University of California.
 * 
 * Use and reproduction of this software are granted  in  accordance  with
 * the terms and conditions specified in  the  Berkeley  Software  License
 * Agreement (in particular, this entails acknowledgement of the programs'
 * source, and inclusion of this notice) with the additional understanding
 * that  all  recipients  should regard themselves as participants  in  an
 * ongoing  research  project and hence should  feel  obligated  to report
 * their  experiences (good or bad) with these elementary function  codes,
 * using "sendbug 4bsd-bugs@BERKELEY", to the authors.
 */

#ifndef lint
static char sccsid[] =
"@(#)exp.c	4.3 (Berkeley) 8/21/85; 1.2 (ucb.elefunt) %G%";
#endif not lint

/* EXP(X)
 * RETURN THE EXPONENTIAL OF X
 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
 * CODED IN C BY K.C. NG, 1/19/85; 
 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85.
 *
 * Required system supported functions:
 *	scalb(x,n)	
 *	copysign(x,y)	
 *	finite(x)
 *
 * Kernel function:
 *	exp__E(x,c)
 *
 * Method:
 *	1. Argument Reduction: given the input x, find r and integer k such 
 *	   that
 *	                   x = k*ln2 + r,  |r| <= 0.5*ln2 .  
 *	   r will be represented as r := z+c for better accuracy.
 *
 *	2. Compute expm1(r)=exp(r)-1 by 
 *
 *			expm1(r=z+c) := z + exp__E(z,r)
 *
 *	3. exp(x) = 2^k * ( expm1(r) + 1 ).
 *
 * Special cases:
 *	exp(INF) is INF, exp(NaN) is NaN;
 *	exp(-INF)=  0;
 *	for finite argument, only exp(0)=1 is exact.
 *
 * Accuracy:
 *	exp(x) returns the exponential of x nearly rounded. In a test run
 *	with 1,156,000 random arguments on a VAX, the maximum observed
 *	error was .768 ulps (units in the last place).
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following constants.
 * The decimal values may be used, provided that the compiler will convert
 * from decimal to binary accurately enough to produce the hexadecimal values
 * shown.
 */

#ifdef VAX	/* VAX D format */
/* double static */
/* ln2hi  =  6.9314718055829871446E-1    , Hex  2^  0   *  .B17217F7D00000 */
/* ln2lo  =  1.6465949582897081279E-12   , Hex  2^-39   *  .E7BCD5E4F1D9CC */
/* lnhuge =  9.4961163736712506989E1     , Hex  2^  7   *  .BDEC1DA73E9010 */
/* lntiny = -9.5654310917272452386E1     , Hex  2^  7   * -.BF4F01D72E33AF */
/* invln2 =  1.4426950408889634148E0     ; Hex  2^  1   *  .B8AA3B295C17F1 */
static long     ln2hix[] = { 0x72174031, 0x0000f7d0};
static long     ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1};
static long    lnhugex[] = { 0xec1d43bd, 0x9010a73e};
static long    lntinyx[] = { 0x4f01c3bf, 0x33afd72e};
static long    invln2x[] = { 0xaa3b40b8, 0x17f1295c};
#define    ln2hi    (*(double*)ln2hix)
#define    ln2lo    (*(double*)ln2lox)
#define   lnhuge    (*(double*)lnhugex)
#define   lntiny    (*(double*)lntinyx)
#define   invln2    (*(double*)invln2x)
#else	/* IEEE double */
double static
ln2hi  =  6.9314718036912381649E-1    , /*Hex  2^ -1   *  1.62E42FEE00000 */
ln2lo  =  1.9082149292705877000E-10   , /*Hex  2^-33   *  1.A39EF35793C76 */
lnhuge =  7.1602103751842355450E2     , /*Hex  2^  9   *  1.6602B15B7ECF2 */
lntiny = -7.5137154372698068983E2     , /*Hex  2^  9   * -1.77AF8EBEAE354 */
invln2 =  1.4426950408889633870E0     ; /*Hex  2^  0   *  1.71547652B82FE */
#endif

double exp(x)
double x;
{
	double scalb(), copysign(), exp__E(), z,hi,lo,c;
	int k,finite();

#ifndef VAX
	if(x!=x) return(x);	/* x is NaN */
#endif
	if( x <= lnhuge ) {
		if( x >= lntiny ) {

		    /* argument reduction : x --> x - k*ln2 */

			k=invln2*x+copysign(0.5,x);	/* k=NINT(x/ln2) */

			/* express x-k*ln2 as z+c */
			hi=x-k*ln2hi;
			z=hi-(lo=k*ln2lo);
			c=(hi-z)-lo;

		    /* return 2^k*[expm1(x) + 1]  */
			z += exp__E(z,c);
			return (scalb(z+1.0,k));  
		}
		/* end of x > lntiny */

		else 
		     /* exp(-big#) underflows to zero */
		     if(finite(x))  return(scalb(1.0,-5000));

		     /* exp(-INF) is zero */
		     else return(0.0);
	}
	/* end of x < lnhuge */

	else 
	/* exp(INF) is INF, exp(+big#) overflows to INF */
	    return( finite(x) ?  scalb(1.0,5000)  : x);
}
Commit	Line	Data
f28ff572 ZAL	1	/*
	2	* Copyright (c) 1985 Regents of the University of California.
	3	*
	4	* Use and reproduction of this software are granted in accordance with
	5	* the terms and conditions specified in the Berkeley Software License
	6	* Agreement (in particular, this entails acknowledgement of the programs'
	7	* source, and inclusion of this notice) with the additional understanding
	8	* that all recipients should regard themselves as participants in an
	9	* ongoing research project and hence should feel obligated to report
	10	* their experiences (good or bad) with these elementary function codes,
	11	* using "sendbug 4bsd-bugs@BERKELEY", to the authors.
	12	*/
	13
	14	#ifndef lint
a62df508 GK	15	static char sccsid[] =
a62df508 GK	16	"@(#)exp.c 4.3 (Berkeley) 8/21/85; 1.2 (ucb.elefunt) %G%";
f28ff572 ZAL	17	#endif not lint
	18
	19	/* EXP(X)
	20	* RETURN THE EXPONENTIAL OF X
	21	* DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
	22	* CODED IN C BY K.C. NG, 1/19/85;
	23	* REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85.
	24	*
	25	* Required system supported functions:
	26	* scalb(x,n)
	27	* copysign(x,y)
	28	* finite(x)
	29	*
	30	* Kernel function:
	31	* exp__E(x,c)
	32	*
	33	* Method:
	34	* 1. Argument Reduction: given the input x, find r and integer k such
	35	* that
	36	* x = kln2 + r, \|r\| <= 0.5ln2 .
	37	* r will be represented as r := z+c for better accuracy.
	38	*
	39	* 2. Compute expm1(r)=exp(r)-1 by
	40	*
	41	* expm1(r=z+c) := z + exp__E(z,r)
	42	*
	43	* 3. exp(x) = 2^k * ( expm1(r) + 1 ).
	44	*
	45	* Special cases:
	46	* exp(INF) is INF, exp(NaN) is NaN;
	47	* exp(-INF)= 0;
	48	* for finite argument, only exp(0)=1 is exact.
	49	*
	50	* Accuracy:
	51	* exp(x) returns the exponential of x nearly rounded. In a test run
	52	* with 1,156,000 random arguments on a VAX, the maximum observed
	53	* error was .768 ulps (units in the last place).
	54	*
	55	* Constants:
	56	* The hexadecimal values are the intended ones for the following constants.
	57	* The decimal values may be used, provided that the compiler will convert
	58	* from decimal to binary accurately enough to produce the hexadecimal values
	59	* shown.
	60	*/
	61
	62	#ifdef VAX /* VAX D format */
	63	/* double static */
	64	/* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
	65	/* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
	66	/* lnhuge = 9.4961163736712506989E1 , Hex 2^ 7 * .BDEC1DA73E9010 */
	67	/* lntiny = -9.5654310917272452386E1 , Hex 2^ 7 * -.BF4F01D72E33AF */
	68	/* invln2 = 1.4426950408889634148E0 ; Hex 2^ 1 * .B8AA3B295C17F1 */
	69	static long ln2hix[] = { 0x72174031, 0x0000f7d0};
	70	static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1};
	71	static long lnhugex[] = { 0xec1d43bd, 0x9010a73e};
	72	static long lntinyx[] = { 0x4f01c3bf, 0x33afd72e};
	73	static long invln2x[] = { 0xaa3b40b8, 0x17f1295c};
	74	#define ln2hi ((double)ln2hix)
	75	#define ln2lo ((double)ln2lox)
	76	#define lnhuge ((double)lnhugex)
	77	#define lntiny ((double)lntinyx)
	78	#define invln2 ((double)invln2x)
	79	#else /* IEEE double */
	80	double static
81	ln2hi = 6.9314718036912381649E-1 , /Hex 2^ -1 1.62E42FEE00000 */
82	ln2lo = 1.9082149292705877000E-10 , /Hex 2^-33 1.A39EF35793C76 */
83	lnhuge = 7.1602103751842355450E2 , /Hex 2^ 9 1.6602B15B7ECF2 */
84	lntiny = -7.5137154372698068983E2 , /Hex 2^ 9 -1.77AF8EBEAE354 */
85	invln2 = 1.4426950408889633870E0 ; /Hex 2^ 0 1.71547652B82FE */
86	#endif
87
88	double exp(x)
89	double x;
90	{
91	double scalb(), copysign(), exp__E(), z,hi,lo,c;
92	int k,finite();
93
94	#ifndef VAX
95	if(x!=x) return(x); /* x is NaN */
96	#endif
97	if( x <= lnhuge ) {
98	if( x >= lntiny ) {
99
100	/* argument reduction : x --> x - kln2 /
101
102	k=invln2x+copysign(0.5,x); / k=NINT(x/ln2) */
103
104	/* express x-kln2 as z+c /
105	hi=x-k*ln2hi;
106	z=hi-(lo=k*ln2lo);
107	c=(hi-z)-lo;
108
109	/* return 2^k[expm1(x) + 1] /
110	z += exp__E(z,c);
111	return (scalb(z+1.0,k));
112	}
113	/* end of x > lntiny */
114
115	else
116	/* exp(-big#) underflows to zero */
117	if(finite(x)) return(scalb(1.0,-5000));
118
119	/* exp(-INF) is zero */
120	else return(0.0);
121	}
122	/* end of x < lnhuge */
123
124	else
125	/* exp(INF) is INF, exp(+big#) overflows to INF */
126	return( finite(x) ? scalb(1.0,5000) : x);
127	}